The main result of the paper is a refinement of Xia's (1997) bound on the Kantorovich distance between distributions of a Bernoulli point process and an approximating Poisson process. In particular, we show that the distance between distributions of a Bernoulli point process and the Poisson process with the same mean measure has the order of the total variation distance between the laws of the total masses of these processes.

Mi (2002) recently considered a two-dimensional optimization problem for the optimal age-replacement policy and the optimal work size. In order to find (y ∗,T ∗), Mi (2002) found the optimal age-replacement policy T ∗(y) for each fixed work size y, and then searched for the optimal work size y ∗. When applying this approach, for each fixed work size y, Mi (2002) obtained the bounds for T ∗(y). However, no bound for the optimal work size y ∗ was derived. In this note, the results on the upper bound for the optimal work size y ∗ are given.

The notion of minimal repair is generalized to the case when the lifetime distribution function is a continuous or a discrete mixture of distributions (heterogeneous population). The statistical (black box) minimal repair and the minimal repair based on information just before the failure of an object are considered. The corresponding stochastic intensities are defined and analyzed for the point processes generated by both types of minimal repair. Some generalizations are discussed. Several simple examples are considered.

For the M[X]/G/1 queueing model with a general exhaustive-service vacation policy, it has been proved that the Laplace-Stieltjes transform (LST) of the steady-state distribution function of the waiting time of a customer arriving while the server is active is the product of the corresponding LST in the bulk arrival model with unremovable server and another LST. The expression given for the latter, however, is valid only under the assumption that the number of groups arriving in an inactive phase is independent of the sizes of the groups. We here give an expression which holds in the general case. For the N-policy case, we also give an expression for the LST of the steady-state distribution function of the waiting time of a customer arriving while the server is inactive.

We show that the critical probability for the frog model on a graph is not a monotonic function of the graph. This answers a question of Alves, Machado and Popov. The nonmonotonicity is unexpected as the frog model is a percolation model.